Tragedy of Schrödinger and His Equation

ChatGPT informs (see also previous post) us that the common view of modern physicists is that the physics of quantum mechanics is weird, which can be seen as the desperation underlying the crisis of fundamental physics commonly recognised. See also previous post.

Let us seek the origin of the idea that the accepted form of fundamental physics, referred below to as Standard Quantum Mechanics stdQM, is weird and as such may not qualify as rational science. 

StdQM was born in early 1926 when Erwin Schrödinger during a happy vacation in the Alps with his girlfriend formulated a mathematical model of the Hydrogen atom consisting of a positive kernel surrounded by negative charge distribution $\psi (x)$ expressing balance between two forms of energy:

• $E_C = -\int\frac{\psi^2(x)}{\vert x\vert}dx$                  Coulomb potential energy 
• $E_K =\frac{1}{2}\int \vert\nabla\psi (x)\vert^2dx$   charge distribution energy ("kinetic energy")

where $x$ is a Euclidean space coordinate with the kernel positioned at $x=0$. Schrödinger also formulated a version of the model depending on a time coordinate, named the Schrödinger equation forming the basis quantum mechanics. The ground state of the Hydrogen atom corresponds to a minimum of the total energy $E_C+E_K$, with the charge distribution as close as possible to the kernel at a gradient cost. This model has the form of classical continuum mechanics as a model of an elastic body in a potential well, very well understood.

Schrödinger's model exactly captured the spectrum of the Hydrogen atom and so was an amazing success only comparable to Newton's law of gravitation, which propelled Schrödinger to fame. 

Schrödinger then faced the challenge to generalise to atoms with more than one electron, starting with Helium with two electrons. A formal generalisation presented itself in terms of a wave function $\psi (x,y)$ depending on two Euclidean space coordinates $x$ and $y$ thus on altogether 6 spatial dimensions, which was first rejected by Schrödinger, because such a function can only describe something beyond real physical space in 3 dimensions. In any case, the easy catch of a formal generalisation was irresistible and so the Schrödinger equation for an atom with $N>1$ electrons of stdQM was formulated as a linear equation for a wave function depending on $3N$ spatial dimensions. 

Combined with a statistical interpretation of the multi-d wave function supplied by Born, this model serves as the foundation of stdQM according to the Copenhagen Interpretation.  

But Schrödinger was not happy with his equation in this form as he recalled in  in Chapter 1 of Thee Interpretation of Quantum Mechanics (Dublin Seminars 1949-1955):

• Let me say at the outset, that in this discourse, I am opposing not a few special statements of quantum mechanics held today, I am opposing as it were the whole of it, I am opposing its basic views that have been shaped 25 years ago, when Max Born put forward his probability interpretation, which was accepted by almost everybody.
• The view I am opposing is so widely accepted, without ever being questioned, that I would have some difficulties in making you believe that I really, really consider it inadequate and wish to abandon it.

In short, Schrödinger could not survive within stdQM and so gradually drifted away into other forms of physics. From inside to outside. A tragedy. Einstein shared the same fate when questioning quantum mechanics as statistics. There are critics still alive such as Penrose but discussion can only take place outside physics departments. Physics is settled, in crisis.

Suppose we do not fall for the temptation of an easy catch in terms of a wave function for the Helium atom $\psi (x,y)$ depending on two 3-d spatial variables $x$ and $y$, and instead seek a wave function as a sum of two one-electron wave functions

• $\psi_1(x) + \psi_2(x)$ 

depending on a common 3-d variable $x$, with $\psi_1(x)$ representing the charge density of one of the electrons of Helium and $\psi_2(x)$ the other, assuming the densities do not overlap. This is a very natural Ansatz, and it would be very interesting to know if Schrödinger ever tried this. 

In any case, if you do try then you come with a different form of Schrödinger equations described as RealQM.  I am sure Schrödinger would have been very happy with this model, since it has a direct deterministic physical interpretation in terms of classical continuum mechanics and gives very good agreement with observations. 

It is possible that with RealQM instead of stdQM much of the crisis of fundamental physics from being weird, will disappear. Ready to try?

If a physical theory still after 100 years of conception, appears as being very strange and incomprehensible, then something must be wrong...